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Mirrors > Home > ILE Home > Th. List > nn0ge2m1nn | GIF version |
Description: If a nonnegative integer is greater than or equal to two, the integer decreased by 1 is a positive integer. (Contributed by Alexander van der Vekens, 1-Aug-2018.) (Revised by AV, 4-Jan-2020.) |
Ref | Expression |
---|---|
nn0ge2m1nn | ⊢ ((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → (𝑁 − 1) ∈ ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 108 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → 𝑁 ∈ ℕ0) | |
2 | 1red 7805 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → 1 ∈ ℝ) | |
3 | 2re 8814 | . . . . . . . . 9 ⊢ 2 ∈ ℝ | |
4 | 3 | a1i 9 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → 2 ∈ ℝ) |
5 | nn0re 9010 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ) | |
6 | 2, 4, 5 | 3jca 1162 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (1 ∈ ℝ ∧ 2 ∈ ℝ ∧ 𝑁 ∈ ℝ)) |
7 | 6 | adantr 274 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → (1 ∈ ℝ ∧ 2 ∈ ℝ ∧ 𝑁 ∈ ℝ)) |
8 | simpr 109 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → 2 ≤ 𝑁) | |
9 | 1lt2 8913 | . . . . . . 7 ⊢ 1 < 2 | |
10 | 8, 9 | jctil 310 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → (1 < 2 ∧ 2 ≤ 𝑁)) |
11 | ltleletr 7870 | . . . . . 6 ⊢ ((1 ∈ ℝ ∧ 2 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((1 < 2 ∧ 2 ≤ 𝑁) → 1 ≤ 𝑁)) | |
12 | 7, 10, 11 | sylc 62 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → 1 ≤ 𝑁) |
13 | elnnnn0c 9046 | . . . . 5 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁)) | |
14 | 1, 12, 13 | sylanbrc 414 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → 𝑁 ∈ ℕ) |
15 | nn1m1nn 8762 | . . . 4 ⊢ (𝑁 ∈ ℕ → (𝑁 = 1 ∨ (𝑁 − 1) ∈ ℕ)) | |
16 | 14, 15 | syl 14 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → (𝑁 = 1 ∨ (𝑁 − 1) ∈ ℕ)) |
17 | 1re 7789 | . . . . . . . . . . 11 ⊢ 1 ∈ ℝ | |
18 | 3, 17 | lenlti 7888 | . . . . . . . . . 10 ⊢ (2 ≤ 1 ↔ ¬ 1 < 2) |
19 | 18 | biimpi 119 | . . . . . . . . 9 ⊢ (2 ≤ 1 → ¬ 1 < 2) |
20 | 9, 19 | mt2 630 | . . . . . . . 8 ⊢ ¬ 2 ≤ 1 |
21 | breq2 3941 | . . . . . . . 8 ⊢ (𝑁 = 1 → (2 ≤ 𝑁 ↔ 2 ≤ 1)) | |
22 | 20, 21 | mtbiri 665 | . . . . . . 7 ⊢ (𝑁 = 1 → ¬ 2 ≤ 𝑁) |
23 | 22 | pm2.21d 609 | . . . . . 6 ⊢ (𝑁 = 1 → (2 ≤ 𝑁 → (𝑁 − 1) ∈ ℕ)) |
24 | 23 | com12 30 | . . . . 5 ⊢ (2 ≤ 𝑁 → (𝑁 = 1 → (𝑁 − 1) ∈ ℕ)) |
25 | 24 | adantl 275 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → (𝑁 = 1 → (𝑁 − 1) ∈ ℕ)) |
26 | 25 | orim1d 777 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → ((𝑁 = 1 ∨ (𝑁 − 1) ∈ ℕ) → ((𝑁 − 1) ∈ ℕ ∨ (𝑁 − 1) ∈ ℕ))) |
27 | 16, 26 | mpd 13 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → ((𝑁 − 1) ∈ ℕ ∨ (𝑁 − 1) ∈ ℕ)) |
28 | oridm 747 | . 2 ⊢ (((𝑁 − 1) ∈ ℕ ∨ (𝑁 − 1) ∈ ℕ) ↔ (𝑁 − 1) ∈ ℕ) | |
29 | 27, 28 | sylib 121 | 1 ⊢ ((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → (𝑁 − 1) ∈ ℕ) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∨ wo 698 ∧ w3a 963 = wceq 1332 ∈ wcel 1481 class class class wbr 3937 (class class class)co 5782 ℝcr 7643 1c1 7645 < clt 7824 ≤ cle 7825 − cmin 7957 ℕcn 8744 2c2 8795 ℕ0cn0 9001 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-addcom 7744 ax-addass 7746 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-0id 7752 ax-rnegex 7753 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-ltadd 7760 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-br 3938 df-opab 3998 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-iota 5096 df-fun 5133 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-inn 8745 df-2 8803 df-n0 9002 |
This theorem is referenced by: nn0ge2m1nn0 9062 |
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